Download Aggregation Functions: A Guide for Practitioners by Gleb Beliakov PDF
By Gleb Beliakov
Aggregation of knowledge is of basic significance within the development of data dependent structures in a variety of domain names, starting from drugs, economics, and engineering to decision-making approaches, man made intelligence, robotics, and laptop studying. This ebook offers a huge advent into the subject of aggregation features, and gives a concise account of the homes and the most periods of such features, together with classical capacity, medians, ordered weighted averaging features, Choquet and Sugeno integrals, triangular norms, conorms and copulas, uninorms, nullnorms, and symmetric sums. It additionally offers a few state of the art options, many graphical illustrations and new interpolatory aggregation services. a specific consciousness is paid to id and development of aggregation capabilities from program particular requisites and empirical information. This e-book presents scientists, IT experts and procedure architects with a self-contained easy-to-use advisor, in addition to examples of computing device code and a software program package deal. it's going to facilitate development of determination aid, specialist, recommender, keep an eye on and lots of different clever systems.
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Additional info for Aggregation Functions: A Guide for Practitioners
The equality f (x1 , . . , xi−1 , a, xi+1 , . . , xn ) = 1, can hold for some x < 1 with a at any position. The interpretation is similar: the value of any inputs larger than a can make the output f (x) = 1, even if none of the inputs is actually 1. On the other hand, if b is not a one divisor, then the output cannot be one if all the inputs are no larger than b. The following property is useful for construction of n-ary aggregation functions from a single two-variable function. 38 (Associativity).
Xn + λ) ∈ [0, 1]n and f (x1 , . . , xn ) + λ ∈ [0, 1]. 46 (Homogeneity). An aggregation function f : [0, 1]n → [0, 1] is homogeneous if for all λ ∈ [0, 1] and for all (x1 , . . , xn ) ∈ [0, 1]n it is f (λx1 , . . , λxn ) = λf (x1 , . . , xn ). Aggregation functions which are both shift-invariant and homogeneous are known as linear aggregation functions. Note that, due to the boundary conditions f (0, . . , 0) = 0 and f (1, . . 12) they can only be found among averaging functions. A prototypical example of a linear aggregation function is the arithmetic mean.
Wn , that are non-negative and add to one is not symmetric, n f (x) = wi xi = w1 x1 + w2 x2 + . . + wn xn . i=1 Permutation of arguments is very important in aggregation, as it helps express symmetry, as well as to deﬁne other concepts. A permutation of (1, 2 . . , 5) is just a tuple like (5, 3, 2, 1, 4). There are n! = 1 × 2 × 3 × . . × n possible permutations of (1, 2, . . , n). We will denote a vector whose components are arranged in the order given by a permutation P by xP = (xP (1) , xP (2) , .